1. TWO CURVES, ONE PRICE Pricing & Hedging Interest Rate Derivatives Using Different Yield Curves For Discounting and ForwardingMarco Bianchetti Intesa San Paolo Bank, Risk Management Market Risk Pricing & Financial Modelling marco.bianchetti intesasanpaolo.com Quant Congress Europe 2009 London,3-5 Nov. 2009

2. Summary1. Context and Market PracticesSingle-Curve Pricing & Hedging Interest-Rate DerivativesFrom Single to Double-Curve Paradigm 2. Double-Curve Framework, No Arbitrage and Forward BasisGeneral AssumptionsDouble-Curve Pricing ProcedureNo Arbitrage and Forward Basis 3. Foreign-Currency Analogy, No Arbitrage and Quanto AdjustmentSpot and Forward Exchange ratesQuanto Adjustment 4. Double-Curve Pricing & Hedging Interest Rate DerivativesPricing> FRAs, Swaps, Caps/Floors/SwaptionsHedging 5. No Arbitrage and Counterparty Risk 6. Other Approaches 7. Conclusions and ReferencesTwo Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 2 3. 1: Context & Market Practices:Single-Curve Pricing & Hedging IR DerivativesPre credit-crunch single curve market practice: select a single set of the most convenient (e.g. liquid) vanilla interest rateinstruments traded on the market with increasing maturities; for instance, avery common choice in the EUR market is a combination of short-termEUR deposit, medium-term FRA/Futures on Euribor3M and medium-long-term swaps on Euribor6M;build a single yield curve C using the selected instruments plus a set ofbootstrapping rules (e.g. pillars, priorities, interpolation, etc.);Compute, on the same curve C, forward rates, cashflows, discountfactors and work out the prices by summing up the discounted cashflows;compute the delta sensitivity and hedge the resulting delta risk using thesuggested amounts (hedge ratios) of the same set of vanillas.Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 3 4. 1: Context & Market Practices:Market Evolution80EUR Basis swaps1M1M vs vs 3M 6MBasis swaps (single701M3M vs vs 12M 6M currency) as 2 swaps:603M vs12M6M vs12M basis spread (bps)5040 1. Euribor 3MT vs RTM330 2. Euribor 6MT vs RTM6 3. BasisTM 6M = RTM RTM 33 620100 1Y 2Y 3Y4Y 5Y6Y 7Y8Y 9Y 10Y 11Y 12Y 15Y 20Y25Y30YQuotations as of 16 Feb. 2009 (source: Reuters ICAPEUROBASIS)Other market evidences:the divergence between deposit (Euribor based) and OIS (OvernightIndexed Swaps, Eonia based) rates;The divergence between FRA rates and the corresponding forward ratesimplied by consecutive deposits (see e.g. refs. [2], [6], [7]. Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 4 5. 1: Context & Market Practices:Multiple-Curve Pricing & Hedging IR DerivativesNo marketPost credit-crunch multiple curve market practice:standardEonia? build a single discounting curve Cd using the preferred procedure;build multiple distinct forwarding curves Cf1 Cfn using the preferreddistinct selections of vanilla interest rate instruments each homogeneousin the underlying rate tenor (typically 1M, 3M, 6M, 12M);compute the forward rates with tenor f on the corresponding forwardingcurve Cf and calculate the corresponding cashflows;compute the corresponding discount factors using the discounting curveCd and work out prices by summing the discounted cashflows;compute the delta sensitivity and hedge the resulting delta risk using thesuggested amounts (hedge ratios) of the corresponding set of vanillas. Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009p. 5 6. 1: Context & Market Practices:RationaleApparently similar interest rate instruments with different underlying ratetenors are characterised, in practice, by different liquidity and credit riskpremia, reflecting the different views and interests of the marketplayers. Thinking in terms of more fundamental variables, e.g. a short rate, thecredit crunch has acted as a sort of symmetry breaking mechanism:from a (unstable) situation in which an unique short rate process was ableto model and explain the whole term structure of interest rates of all tenors,towards a sort of market segmentation into sub-areas corresponding toinstruments with different underlying rate tenors, characterised, inprinciple, by distinct dynamics, e.g. different short rate processes. Notice that market segmentation was already present (and wellunderstood) before the credit crunch (see e.g. ref. [3]), but not effectivedue to negligible basis spreads.Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 6 7. 2: Multiple-Curve Framework:General Assumptions 1. There exist multiple different interest rate sub-marketsMx, x = {d,f1 ,,fn} characterized by the same currency and by distinctbank accounts Bx and yield curves C x := {T Px ( t0 ,T ) ,T t0 } , 2. The usual no arbitrage relationPx ( t,T2 ) = Px ( t,T1 ) Px ( t,T1,T2 )holds in each interest rate market Mx. 3. Simple compounded forward rates are defined as usual for tT1< T2 Px ( t,T2 ) 1Px ( t,T1,T2 ) = =, Px ( t,T1 ) 1 + Fx ( t;T1,T2 ) x (T1,T2 )T 4. FRA pricing under Qx 2 forward measure associated to numerairePx ( t,T2 ) FRAx ( t;T1,T2 , K ) = Px ( t,T2 ) x (T1,T2 ) { EQx [ Lx (T1,T2 ) ] K }T2 t = Px ( t,T2 ) x (T1,T2 ) [ Fx ( t;T1,T2 ) K ], Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 7 8. 2: Multiple-Curve Framework:Pricing Procedure 1. assume Cd as the discounting curve and Cf as the forwarding curve; 2. calculate any relevant spot/forward rate on the forwarding curve Cf asPf ( t , Ti 1 ) Pf ( t , Ti ) Ff ( t ;Ti 1 , Ti ) =, t Ti 1 < Ti , f (Ti 1 , Ti ) Pf ( t , Ti ) 3. calculate cashflows ci, i = 1,...,n, as expectations of the i-th coupon payoffTii with respect to the discounting Ti - forward measureQd T Qd ici := c ( t,Ti , i ) = Et[ i ];4. calculate the price at time t by discounting each cashflow ci using thecorresponding discount factor Pd ( t,Ti ) obtained from the discountingcurve Cd and summing, n T ( t,T ) =Qd i P ( t,Ti ) Et [ i ];i =1 5. Price FRAs as{ }T Qd 2 FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) f (T1,T2 ) Et [ Ff (T1 ;T1,T2 ) ] K Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 8 9. 2: Multiple-Curve Framework:No Arbitrage and Forward Basis Classic single-curve no arbitrage relations are broken: for instance, by specifying the subscripts d and f as prescribed above we obtain the two eqs.Pd ( t,T2 ) = Pd ( t,T1 ) Pf ( t,T1,T2 ) ,1 Pf ( t,T2 )Pf ( t,T1,T2 ) == ,1 + Ff ( t;T1,T2 ) f (T1,T2 ) Pf ( t,T1 )that clearly cannot hold at the same time. No arbitrage is recovered by taking into account the forward basis as follows1 1 Pf ( t,T1,T2 ) =:= ,1 + Ff ( t;T1,T2 ) f (T1,T2 )1 + [ Fd ( t;T1,T2 ) + BAfd ( t;T1,T2 ) ] d (T1,T2 )for which we obtain the following static expression in terms of discount factors1 Pf ( t,T1 ) Pd ( t,T1 ) BAfd ( t;T1,T2 ) = .d (T1,T2 ) Pf ( t,T2 ) Pd ( t,T2 ) Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009p. 9 10. 2: Multiple-Curve Framework:Forward Basis Curves80 Forward Basis 0Y-3Y10 Forward Basis 3Y-30Y60 8 640 420 basis points basis points 2 0 0 -20-2 -4-40 1M vs Disc1M vs Disc3M vs Disc -6 3M vs Disc-60 6M vs Disc6M vs Disc12M vs Disc-812M vs Disc-80 -10 May-09 May-10 May-11 Nov-09 Nov-10 Nov-11Aug-09Aug-10Aug-11Feb-09Feb-10Feb-11Feb-12Feb-12 Feb-15Feb-18 Feb-21Feb-24 Feb-27Feb-30 Feb-33Feb-36 Feb-39Forward basis (bps) as of end of day 16 Feb. 2009, daily sampled 3M tenor forward rates calculated on C1M, C3M , C6M , C12M curves against Cd taken as reference curve. Bootstrapping as described in ref. [2]. The richer term structure of the forward basis curves provides a sensitive indicator of the tiny, but observable, statical differences between different interest rate market sub-areas in the post credit crunch interest rate world, and a tool to assess the degree of liquidity and credit issues in interest rate derivatives' prices. Provided thatTwo Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009p. 10 11. 2: Multiple-Curve Framework:Bad Curves6% 3M curves 0Y-30Y4 Forward Basis 3Y-30Y25% 0 basis points4% -2 3% -4-61M vs Disc2%3M vs Disczero rates -8 6M vs Discforward rates 12M vs Disc1%-10 Feb-09Feb-11 Feb-13Feb-15 Feb-17Feb-19 Feb-21Feb-23 Feb-25Feb-27 Feb-29Feb-31 Feb-33Feb-35 Feb-37Feb-39Feb-12 Feb-15Feb-18 Feb-21Feb-24 Feb-27Feb-30 Feb-33Feb-36 Feb-39 Left: 3M zero rates (red dashed line) and forward rates (blue continuous line). Right: forward basis. Linear interpolation on zero rates has been used. Numerical results from QuantLib (www.quantlib.org).smooth yield curves are usedNon-smooth bootstrapping techniques, e.g. linear interpolation on zero rates (still a diffused market practice), produce zero curves with no apparent problems, but ugly forward curves with a sagsaw shape inducing, in turn, strong and unnatural oscillations in the forward basis (see [2]).Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009p. 11 12. 3: Foreign Currency Analogy: Spot and Forward Exchange RatesA second issue regarding no arbitrage arises in the double-curve framework: { [ Ff (T1 ;T1,T2 ) ] K }TQd 2FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) f (T1,T2 ) Et Pd ( t,T2 ) f (T1,T2 ) [ Ff (T1 ;T1,T2 ) K ]1. Double-curve-double-currency: d = domestic, f = foreign c d ( t ) = x fd ( t ) c f ( t ) ,X fd ( t , T ) Pd ( t , T ) = x fd ( t ) P f ( t , T ),x fd ( t 0 ) = x fd , 0. 2. Double-curve-single-currency: d = discounting, f=forwarding cd ( t ) = x fd ( t ) c f ( t ) , X fd ( t , T ) Pd ( t , T ) = x fd ( t ) Pf ( t , T ) , x fd ( t 0 ) = 1. Picture of no arbitrage definition of the forward exchange rate. Circuitation (round trip) no money is created or destructed.Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 12 13. 3: Foreign Currency Analogy:Quanto Adjustment 1. Assume a lognormal martingale dynamic for the Cf(foreign) forward rate dFf ( t ;T1,T2 )= f ( t ) dW fT2 ( t ) , Q T2 Pf ( t ,T2 ) C f ; f Ff ( t ;T1,T2 ) 2. since x fd ( t ) Pf ( t ,T ) is the price at time t of a Cd (domestic) tradableasset, the forward exchange rate must be a martingale processdX fd ( t ,T2 )TT= X ( t ) dWX 2 ( t ) , Qd 2 Pd ( t ,T2 ) C d , X fd ( t ,T2 )withdW fT2 ( t ) dWX 2 ( t ) = fX ( t ) dt ; T 3. by changing numeraire from Cf toCdwe obtain the modified dynamicdFf ( t ;T1,T2 ) = f ( t ) dt + f ( t ) dW fT2 ( t ) , Qd 2 Pd ( t ,T2 ) C d , T Ff ( t ;T1,T2 ) f ( t ) = f ( t ) X ( t ) fX ( t ) ; 4. and the modified expectation including the (additive) quanto-adjustment TQd 2Et [ L f (T1,T2 ) ] = Ff ( t ;T1,T2 ) + QAfd ( t ;T1, f , X , fX ),T1QAfd ( t ;T1, f , X , fX ) = Ff ( t ;T1,T2 ) exp t f ( s ) ds 1 . Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 13 14. 4: Pricing & Hedging IR Derivatives:Pricing Plain Vanillas [1] 1. FRA: FRA ( t;T1,T2 , K ) = Pd ( t,T2 ) f (T1,T2 ) [ Ff ( t;T1,T2 ) + QAfd ( t,T1, f , X , fX ) K ] m 2. Swaps:Swap ( t;T , S, K ) = Pd ( t, S j ) d ( S j 1, S j )K j n j =1+ Pd ( t, ST ) f (Tj 1,Tj )[ Ff ( t;Ti 1,Ti ) + QAfd ( t,Ti 1, f ,i , X ,i , fX ,i ) ].i =1 n 3. Caps/Floors: CF ( t;T , K , ) = Pd ( t,Ti ) d (Ti 1,Ti )i =1 Black [ Ff ( t;Ti 1,Ti ) + QAfd ( t,Ti 1, f ,i , X ,i , fX ,i ), Ki , f ,i , v f ,i , i ],4. Swaptions: Swaption ( t;T , S, K , ) = Ad ( t, S ) Black [ S f ( t;T , S ) + QAfd ( t,T , S, f , Y , fY ), K , f , v f , ]. Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 14 15. 4: Pricing & Hedging IR Derivatives:Pricing Plain Vanillas [2] 100 Quanto Adjustment (additive)8060 Quanto adj. (bps)40Numerical scenarios for the (additive)20quanto adjustment, corresponding to 0three different combinations of (flat)-20 volatility values as a function of the-40 correlation. The time interval is fixed-60 Sigma_f = 10%, Sigma_X = 2.5%to T1-t=10 years and the forward rate Sigma_f = 20%, Sigma_X = 5%to 3%.-80Sigma_f = 30%, Sigma_X = 10% -100-1.0 -0.8 -0.6 -0.4 -0.2 -0.0 0.2 0.4 0.6 0.8 1.0Correlation We notice that the adjustment may be not negligible. Positive correlation implies negative adjustment, thus lowering the forward rates. The standard market practice, with no quanto adjustment, is thus not arbitrage free. In practice the adjustment depends on market variables not directly quoted on the market, making virtually impossible to set up arbitrage positions and locking today positive gains in the future. Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 15 16. 4: Pricing & Hedging IR Derivatives:Hedging 1. Given any portfolio of interest rate derivatives with price ( t,T , Rmkt ),compute delta risk with respect to both curves Cd and Cf : ( t,T , Rmkt ) = d ( t,T , Rd ) + ( t,T , Rf )mktf mkt Nd ( t,T , Rmkt ) ( t,T , Rmkt )Nf = Rd (Tj )mkt + mkt(Tj ) , j =1 j =1 Rf2. eventually aggregate it on the subset of most liquid market instruments(hedging instruments);3. calculate hedge ratios: ( t,T , Rmkt ) mkt hx , j = x , j , Rx (Tj )mktmktmktx , j ( t ) x , j =, x = f , d.Rx (Tj ) mktTwo Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 16 17. 5: No Arbitrage and Counterparty Risk:A Simple Credit Model (adapted from ref. [6]) Both the forward basis and the quanto adjustment discussed above find a simple financial explanation in terms of counterparty risk. If we identify:Pd(t,T) = default free zero coupon bond,Pf(t,T) = risky zero coupon bond emitted by a risky counterparty formaturity T and with recovery rate Rf ,(t)>t = (random) counterparty default time observed at time t,qd ( t,T ) = EtQd {1[ ( t )>T ] } = default probability after time T expected at time t,we obtain the following expressionsPf ( t,T ) = Pd ( t,T ) R ( t; t,T, Rf ),1 Pd ( t,T1 ) R ( t; t,T1, Rf )Ff ( t;T1,T2 ) = 1 , f (T1,T2 ) Pd ( t,T2 ) R ( t; t,T2, Rf )where:R ( t;T1,T2, Rf ) = Rf + ( 1 Rf ) EtQd [ qd (T1,T2 ) ].Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 17 18. 5: No Arbitrage and Counterparty Risk:A Simple Credit Model [2]If Ld(T1,T2), Lf(T1,T2) are the risk free and the risky Xibor rates underlying the corresponding derivatives, respectively, we obtain: Pd ( t,T1 ) FRAf ( t;T1,T2, K ) = [ 1 + K f (T1,T2 ) ] Pd ( t,T2 ) , R ( t;T1,T2, Rf ) 1Pd ( t,T1 ) R ( t; t,T1, Rf ) BAfd ( t;T1,T2 ) = 1 ,d (T1,T2 ) Pd ( t,T2 ) R ( t; t,T2, Rf )1Pd ( t,T1 ) 1R ( t; t,T1, Rf ) QAfd ( t;T1,T2 ) = , f (T1,T2 ) Pd ( t,T2 ) R ( t;T1,T2, Rf ) R ( t; t,T2, Rf ) That is, the forward basis and the quanto adjustment expressed in terms of risk free zero coupon bonds Pd(t,T) and of the expected recovery rate. Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 18 19. 6: Pros & Cons, Other Approaches:PROs CONs Simple and familiar framework, no Unobservable exchange rate and additional effort, just analogy. parameters. Straightforward interpretation in Plain vanilla prices acquire volatility terms of counterparty risk.and correlation dependence. M. Henrard: ab-initio parsimonious model [5] F. Mercurio: generalised Libor Market Model [6] M. Morini: full credit model [7] F. Kijima et al: DLG model [8]Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 19 20. 7: Conclusions 1. We have reviewed the pre and post credit crunch market practices forpricing & hedging interest rate derivatives. 2. We have shown that in the present double-curve framework standardsingle-curve no arbitrage conditions are broken and can be recoveredtaking into account the forward basis; once a smooth bootstrappingtechnique is used, the richer term structure of the calculated forward basiscurves provides a sensitive indicator of the tiny, but observable, staticaldifferences between different interest rate market sub-areas. 3. Using the foreign-currency analogy we have computed the no arbitragegeneralised double-curve-single-currency market-like pricingexpressions for basic interest rate derivatives, including a quantoadjustment arising from the change of numeraires naturally associated tothe two yield curves. Numerical scenarios show that the quantoadjustment can be non negligible. 4. Both the forward basis and the quanto adjustment have a simpleinterpretation in terms of counterparty risk, using a simple credit modelwith a risk-free and a risky zero coupond bonds.Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 20 21. 7: Main references[1] M. Bianchetti, Two Curves, One Price: Pricing & Hedging Interest Rate Derivatives UsingDifferent Yield Curves for Discounting and Forwarding, (2009), available at SSRN:http://ssrn.com/abstract=1334356. [2] F. Ametrano, M. Bianchetti, Bootstrapping the Illiquidity: Multiple Yield Curves ConstructionFor Market Coherent Forward Rates Estimation, in Modeling Interest Rates: LatestAdvances for Derivatives Pricing, edited by F. Mercurio, Risk Books, 2009. [3] B. Tuckman, P. Porfirio, Interest Rate Parity, Money Market Basis Swaps, and Cross-Currency Basis Swaps, Lehman Brothers, Jun. 2003. [4] W. Boenkost, W. Schmidt, Cross currency swap valuation, working Paper, HfB--BusinessSchool of Finance & Management, May 2005. [5] M. Henrard, The Irony in the Derivatives Discounting - Part II: The Crisis, working paper, Jul.2009, available at SSRN: http://ssrn.com/abstract=1433022. [6] F. Mercurio, "Post Credit Crunch Interest Rates: Formulas and Market Models", workingpaper, Bloomberg, 2009, available at SSRN: http://ssrn.com/abstract=1332205. [7] M. Morini, "Credit Modelling After the Subprime Crisis", Marcus Evans course, 2008. [8] M. Kijima, K. Tanaka, T. Wong, A Multi-Quality Model of Interest Rates, QuantitativeFinance, 2008. [9] D. Brigo, F. Mercurio, "Interest Rate Models - Theory and Practice", 2nd ed., Springer, 2006.Two Curves, One Price - Marco Bianchetti Quant Congress Europe London,3-5 Nov. 2009 p. 21